Abstract

I review the development of numerical evolution codes for general relativity based upon the characteristic initial value problem. Progress is traced from the early stage of 1D feasibility studies to 2D axisymmetric codes that accurately simulate the oscillations and gravitational collapse of relativistic stars and to current 3D codes that provide pieces of a binary black hole spacetime. A prime application of characteristic evolution is to compute waveforms via Cauchy-characteristic matching, which is also reviewed.

Highlights

  • We are in an era in which Einstein’s equations can effectively be considered solved at the local level

  • There is no single code in existence today which purports to be capable of computing the waveform of gravitational radiation emanating from the inspiral and merger of two black holes, the premier problem in classical relativity

  • Most of the effort in numerical relativity has centered about the Cauchy {3 + 1} formalism [226], with the gravitational radiation extracted by perturbative methods based upon introducing an artificial Schwarzschild background in the exterior region [1, 4, 2, 3, 181, 180, 156]

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Summary

Introduction

We are in an era in which Einstein’s equations can effectively be considered solved at the local level. We trace the development of characteristic algorithms from model 1D problems to a 2D axisymmetric code which computes the gravitational radiation from the oscillation and gravitational collapse of a relativistic star and to a 3D code designed to calculate the waveform emitted in the merger to ringdown phase of a binary black hole. Several numerical relativity codes for treating the problem of a neutron star near a black hole have been developed, as described in the Living Review in Relativity on “Numerical Hydrodynamics in General Relativity” by Font [80] Most of these efforts concentrate on Cauchy evolution, the characteristic approach has shown remarkable robustness in dealing with a single black hole or relativistic star. Animations and other material from these studies can be viewed at the web sites of the University of Canberra [217], Louisiana State University [148], Pittsburgh University [218], and Pittsburgh Supercomputing Center [145]

The Characteristic Initial Value Problem
Prototype Characteristic Evolution Codes
Adaptive mesh refinement
The Bondi problem
The conformal-null tetrad approach
Axisymmetric mode coupling
Twisting axisymmetry
The Bondi mass
Geometrical formalism
Numerical methodology
Stability
Accuracy
First versus second differential order
Nonlinear scattering off a Schwarzschild black hole
Black hole in a box
Characteristic treatment of binary black holes
Perturbations of Schwarzschild
Close approximation white hole and black hole waveforms
Fissioning white hole
Nonlinear mode coupling
Cauchy-Characteristic Matching
Computational boundaries
The computational matching strategy
Perturbative matching schemes
Cauchy-characteristic matching for 1D gravitational systems
Cylindrical matching
Spherical matching
Excising 1D black holes
Axisymmetric Cauchy-characteristic matching
Cauchy-characteristic matching for 3D scalar waves
Stable implementation of 3D linearized Cauchy-characteristic matching
The binary black hole inner boundary
Numerical Hydrodynamics on Null Cones
Spherically symmetric hydrodynamic codes
Axisymmetric characteristic hydrodynamic simulations
Three-dimensional characteristic hydrodynamic simulations
Massive particle orbiting a black hole
Computing the radiation field
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